I agree that a ginger grater does a finer job than microplane, but I can't agree with your description of the result. If you grate ginger this way, you get two different things: very finely grated ginger and ginger juice. In some cases you want one or the other, sometimes both.
For this recipe, given that the ginger is supposed to be mixed with some kind of crumb mixture, I'd say that you want the ginger as finely ground as possible, with minimal fibers.
1. If you like ginger a lot, peel and mince the ginger very fine with a knife.
2. If you aren't sure you like so much ginger, grate an inch -- don't bother peeling it first -- with the grater set over a smallish bowl. Squeeze all the grated stuff very hard between your fingers to get all the juice into the bowl. Use the juice only.
3. Peel and microplane the ginger to get a result somewhere in between.
I would recommend against a processor for this small an amount. You'll end up spending an awful lot of time scraping stuff back down into the bowl, and it won't do a great job anyway.
Note that it is unnecessary to peel ginger: just cut off the hard, horny bits where the root was broken or cut and healed itself over. Scrub the peel, and then use like anything else. If the ginger is fresh, the peel will be perfectly edible. If the ginger is getting a bit old and woody, with a dry peel, remove it: all the flavor will have gone. It's only important to peel fresh ginger if you are going to julienne it very fine as a garnish and want it to be beautiful.
As far as picking your ginger, you want something that looks like a tan root. Don't spend extra on young ginger with the little pink-white stalks; try that if you decide you love the dish and want to experiment. Look the tan root all over carefully before you buy: it should be firm, almost rock-hard, with a generally unblemished skin. Some blemishes are reasonable near the tips and where the root has clearly been broken apart, but it should not have a gnarly sort of wood-bark surface. This sort of skin and a distinctly yielding skin are generally signs of old or badly-stored ginger.
Cut ginger stores very well, treated appropriately. Drop the ginger in an unwaxed paper lunchbag. Wrap up, then put the bag in a plastic bag and twist-tie the neck. Place in the refrigerator. Check every now and again, but ginger stored this way should keep for a month or more.